Let's use some numbers as an illustration.
For example,
P = [1 1;1 1]; T = [ 1 2;3 4 ];
L = P*T*P = [10 10;10 10];
X = P*T*T*T*T*T*T*P = [44966 44966;44966 44966];
We only have L and X. The problem is how to get back P and T.
I am working with 2 by 2 matrices, L, X, P and T.
L = PTP -------Eqn 1
X = PTTTTTTP ----------Eqn 2
where L and X are known matrices. P and T matrix are unknown and are to be solved.
The above problem looks simple but are you able to solve it?